There has been known an aplanatic lens, as an aberration-free lens (e.g., see Hiroshi Kubota, “Optics”, 12th edition, Iwanami Shoten Co., Apr. 9, 1986, pp. 282-283).
In a lens with a lens surface 71 shown in FIG. 10, in order to collect a light beam Lb parallel to an optical axis Opa of the lens at a focal point F with a constant light path length, it is necessary to meet a condition of “RF=HF”, where “R” denotes a refraction point in the lens surface 71, “H” denotes a point at the intersection the optical axis Opa with a perpendicular extending downward from the refraction point R (that is, a foot of a perpendicular extending toward the optical axis Opa from the refraction point R), “RF” denotes a light path length between the refraction point R and the focal point F, and “HF” denotes a light path length between the intersection point H and the focal point F. In order to meet the condition of “RF=HF”, it is known that a hyperboloid or an ellipsoid is necessarily adopted as the lens surface 71. Here, in a case where the lens surface 71 is a hyperboloid, when a refraction index of a lens material is denoted by “n” and a back focus of the lens is denoted by “f”, the lens surface 71 is represented by the following Formula (1).
                              [                      Math            .                                                  ⁢            1                    ]                ⁢                                                                                                                                          (                                  z                  -                  c                                )                            2                                      a              2                                -                                                    x                2                            +                              y                2                                                    b              2                                      =        1                            Formula        ⁢                                  ⁢                  (          1          )                    
Here, Formula (1) is obtained when defining a rectangular coordinate system, in which the focal point F of the lens is employed as an origin, a z-axis is specified on the optical axis Opa, a x-axis and a y-axis orthogonal to each other on any place surface orthogonal to the optical axis Opa are specified, and defining a coordinate of an arbitrary point on the lens surface 71 as (x, y, z). Then, a, b and c in Formula (1) are provided by the following Formula (2), Formula (3) and Formula (4), respectively.
                              [                      Math            .                                                  ⁢            2                    ]                ⁢                                                                                      a        =                  f                      n            +            1                                              Formula        ⁢                                  ⁢                  (          2          )                                        b        =                                                            n                -                1                                            n                +                1                                              ⁢          f                                    Formula        ⁢                                  ⁢                  (          3          )                                        c        =                              n                          n              +              1                                ⁢          f                                    Formula        ⁢                                  ⁢                  (          4          )                    
Further, as shown in FIG. 11, there has been known a condenser lens 101, in which a rotation axis C of a hyperboloid 120 that is an emission surface (a second surface) is oblique to a normal line N of a plane surface 110 that is an incident surface (a first surface) so that an angle θ is formed by the rotation axis C and the normal line N (Japanese Examined Patent Publication No. 7-36041). In the condenser lens 101 having the configuration shown in FIG. 11, when a light beam is incident upon the condenser lens 101 with an angle δ formed by the light beam and the rotation axis C, the light beam becomes parallel to the rotation axis C of the hyperboloid 120, inside the condenser lens 101, and then is aplanatically collected on the focal point. F. When a refraction index of the condenser lens 101 is denoted by “n”, the angle δ meets Snell's law, that is, sin(θ+δ)=n*sin θ. In this case, the hyperboloid 120 is represented by the abovementioned Formula (1), when defining a rectangular coordinate system, in which the focal point F is employed as an origin, a z-axis is specified on the rotation axis C of hyperboloid 120, a x-axis and a y-axis orthogonal to each other on any place surface orthogonal to the rotation axis C are specified.
Further, in the abovementioned Japanese Examined Patent Publication, as shown in FIGS. 12A and 12B, there has been proposed the condenser lens 101, which is a Fresnel lens, and in which the rotation axis C shared by hyperboloids 121, 122 and 123 of a second surface is oblique to the plane surface 110 of a first surface in order to suppress occurrence of an off-axis aberration. In this case, the respective hyperboloids 121, 122 and 123 configure lens surfaces.
The above-mentioned Japanese Examined Patent Publication describes that in the Fresnel lens 101 in FIGS. 12A and 12B an angle can be formed between a parallel light beam aplanatically collected on a focal point and a normal line N of the plane surface 110 according to an angle formed by the rotation axis C shared by the hyperboloids 121, 122 and 123 and the plane surface 110. Therefore, in the Fresnel lens 101 in FIGS. 12A and 12B, the occurrence of the off-axis aberration can be suppressed, and light beams from a direction oblique to the normal line N of the plane surface 110 can be effectively collected.
However, in the Fresnel lens 101 configured such that the rotation axis C of the hyperboloids 121, 122 and 123 configuring the emission surface is oblique to the normal line N of the plane surface 110 that is the incident surface, the hyperboloids 121, 122 and 123 are not rotationally symmetric with respect to the normal line N of the plane surface 110. Therefore, the Fresnel lens 101 or a metal mold for the Fresnel lens 101 is difficult to be produced by rotary forming with a lathe or the like.
So, when the Fresnel lens 101 or the metal mold for the Fresnel lens 101 is produced, it is necessary to use a multiaxis control processing machine and form the hyperboloids 121, 122 and 123 or respective curved surfaces by cutting at minute pitches while only a blade edge of a sharp cutting tool (tool) 130 with a nose radius (also referred to as a corner radius) of a few micro-meters is brought into point contact with a workpiece 140, as shown in FIG. 13. The workpiece 140 is a base material for directly forming the Fresnel lens 101, or a base material for forming the metal mold. Therefore, the processing time for producing the aforementioned Fresnel lens 101 or metal mold for the Fresnel lens 101 is increased, and then the cost of the Fresnel lens 101 is increased.
On the other hand, in a case where the cross-sectional shape of each lens surface in the cross-sectional shape including the normal line of the plane surface that is the incident surface of the Fresnel lens is linear, the lens surfaces or the curved surfaces corresponding to the lens surfaces can be formed by cutting while the cutting tool 130 is inclined with respect to the workpiece 140 so as to bring a side surface of a blade into line contact with the workpiece 140, as shown in FIG. 14, thus enabling significant reduction of the processing time. Here, in a Fresnel lens in which the shape of each lens surface in an emission surface is rotationally symmetric by employing a normal line of the incident surface as a rotation axis, it is known that each lens surface is approximated by a side surface of a frustum of cone, thereby enabling the cross-sectional shape of each lens surface to become linear (U.S. Pat. No. 4,787,722).
In the Fresnel lens 101 disclosed in the above-mentioned Japanese Examined Patent Publication and the Fresnel lens disclosed in the above-mentioned US patent, an intended light beam is infrared light, and these two documents disclose that polyethylene is used as a lens material.
Incidentally, in the Fresnel lens in which the shape of each lens surface in an emission surface is rotationally symmetric by employing the normal line of the incident surface as the rotation axis, in the lens surface being approximated by the side surface of the frustum of cone, an off-axis aberration occurs.